General Computer Science; Problems and Solutions for ... - Kwarc

Problem 1.38 (Substitution Application)
Consider the following SML data type of terms:
datatype term = const of string
| var of string
| pair of term * term
| appl of string * term
Constants and variables are represented by a constructor taking their name string, whereas applications
of the form f(t) are constructed from the name string and the argument. Remember
that we use f(a, b) as an abbreviation for f(〈a, b〉). Thus a term f(a, g(x)) is represented as
appl("f",pair(const("a"), appl("g", var("x")))).
With this, we can represent substitutions as lists of elementary substitutions, which are pairs
of type term * string. Thus we can set
type subst = term * string list
and represent a substitution σ = [(f(a))/x], [b/y] as [(appl("f", const("a")), "x"), (const("b"), "y")].
Of course we may not allow ambiguous substitutions which contain duplicate strings.
Write an SML function substApply for the substitution application operation, i.e. substApply
takes a substitution σ and a term A as arguments and returns the term σ(A) if σ is unambiguous
and raises an exception otherwise.
Make sure that your function applies substitutions in a parallel way, i.e. that [y/x], [x/z](f(z)) =
f(x).
Solution:
exception ambiguous_substitution
local
fun sa(s,const(str)) = const(str)
| sa(s,pair(t1,t2)) = pair(sa(s,t1),sa(s,t2))
| sa(s,appl(fs,t1)) = appl(fs,sa(s,t1))
| sa(nil,var(str)) = var(str)
| sa((t,x)::L,var(str)) = if str = x then t else sa(L,var(str))
fun ambiguous = ...
in
fun substApply (s,t) = if ambiguous(s)
then raise ambiguous_substitution
else sa(s,t)
end
or
(* (C) by Anna Michalska *)
datatype term = const of string
| var of string
| pair of term * term
| appl of string * term;
type subst = (term * string) list;
exception ania;
fun comparing1 ((x1,x2), []) = true | comparing1 ((x1,x2), hd::tl) = if
hd=x2 then false else comparing1 ((x1,x2),tl);
fun comparing2([],_)=true | comparing2 ((x3,x4)::t,tl) = if (comparing1
((x3,x4),tl)) then comparing2 (t,x4::tl) else raise ania;
fun tab (a,[]) = var(a)
| tab (a, (a1,a2)::tl) = if (a=a2) then a1 else tab(a,tl);
fun substApply_r (appl(a,b),subst_in) = appl(a,substApply_r(b,subst_in))
|substApply_r (pair(a,b),subst_in) =
pair(substApply_r(a,subst_in),substApply_r(b,subst_in))
|substApply_r (var(a),subst_in) = tab(a,subst_in)
|substApply_r (const(x),subst_in) = const(x);
39
30pt